EllDistrEst: Nonparametric estimation of the density generator of an...
In ElliptCopulas: Inference of Elliptical Distributions and Copulas

EllDistrEst

R Documentation

Nonparametric estimation of the density generator of an elliptical distribution

Description

This function uses Liebscher's algorithm to estimate the density generator of an elliptical distribution by kernel smoothing. A continuous elliptical distribution has a density of the form

f_X(x) = {|\Sigma|}^{-1/2} g\left( (x-\mu)^\top \, \Sigma^{-1} \, (x-\mu) \right),

where x \in \mathbb{R}^d, \mu \in \mathbb{R}^d is the mean, \Sigma is a d \times d positive-definite matrix and a function g: \mathbb{R}_+ \rightarrow \mathbb{R}_+, called the density generator of X. The goal is to estimate g at some point \xi, by

\widehat{g}_{n,h,a}(\xi) := \dfrac{\xi^{\frac{-d+2}{2}} \psi_a'(\xi)}{n h s_d} \sum_{i=1}^n K\left( \dfrac{ \psi_a(\xi) - \psi_a(\xi_i) }{h} \right) + K\left( \dfrac{ \psi_a(\xi) + \psi_a(\xi_i) }{h} \right),

where s_d := \pi^{d/2} / \Gamma(d/2), \Gamma is the Gamma function, h and a are tuning parameters (respectively the bandwidth and a parameter controlling the bias at \xi = 0), \psi_a(\xi) := -a + (a^{d/2} + \xi^{d/2})^{2/d}, \xi \in \mathbb{R}, K is a kernel function and \xi_i := (X_i - \mu)^\top \, \Sigma^{-1} \, (X_i - \mu), for a sample X_1, \dots, X_n.

Usage

EllDistrEst(
  X,
  mu = 0,
  Sigma_m1 = diag(d),
  grid,
  h,
  Kernel = "epanechnikov",
  a = 1,
  mpfr = FALSE,
  precBits = 100,
  dopb = TRUE
)

Arguments

`X`	a matrix of size `n \times d`, assumed to be `n` i.i.d. observations (rows) of a `d`-dimensional elliptical distribution.
`mu`	mean of X. This can be the true value or an estimate. It must be a vector of dimension `d`.
`Sigma_m1`	inverse of the covariance matrix of X. This can be the true value or an estimate. It must be a matrix of dimension `d \times d`.
`grid`	grid of values of `\xi` at which we want to estimate the density generator.
`h`	bandwidth of the kernel. Can be either a number or a vector of the size `length(grid)`.
`Kernel`	name of the kernel. Possible choices are `"gaussian"`, `"epanechnikov"`, `"triangular"`.
`a`	tuning parameter to improve the performance at 0. Can be either a number or a vector of the size `length(grid)`. If this is a vector, the code will need to allocate a matrix of size `nrow(X) * length(grid)` which can be prohibitive in some cases.
`mpfr`	if `mpfr = TRUE`, multiple precision floating point is used via the package Rmpfr. This allows for a higher (numerical) accuracy, at the expense of computing time. It is recommended to use this option for higher dimensions.
`precBits`	number of precBits used for floating point precision (only used if `mpfr = TRUE`).
`dopb`	a Boolean value. If `dopb = TRUE`, a progress bar is displayed.

Value

the values of the density generator of the elliptical copula, estimated at each point of the grid.

Author(s)

Alexis Derumigny, Rutger van der Spek

References

Liebscher, E. (2005). A semiparametric density estimator based on elliptical distributions. Journal of Multivariate Analysis, 92(1), 205. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.jmva.2003.09.007")}

The function \psi_a is introduced in Liebscher (2005), Example p.210.

Examples

# Comparison between the estimated and true generator of the Gaussian distribution
X = matrix(rnorm(500*3), ncol = 3)
grid = seq(0,5,by=0.1)
g_3 = EllDistrEst(X = X, grid = grid, a = 0.7, h=0.05)
g_3mpfr = EllDistrEst(X = X, grid = grid, a = 0.7, h=0.05,
                      mpfr = TRUE, precBits = 20)
plot(grid, g_3, type = "l")
lines(grid, exp(-grid/2)/(2*pi)^{3/2}, col = "red")

# In higher dimensions

d = 250
X = matrix(rnorm(500*d), ncol = d)
grid = seq(0, 400, by = 25)
true_g = exp(-grid/2) / (2*pi)^{d/2}

g_d = EllDistrEst(X = X, grid = grid, a = 100, h=40)

g_dmpfr = EllDistrEst(X = X, grid = grid, a = 100, h=40,
                      mpfr = TRUE, precBits = 10000)
ylim = c(min(c(true_g, as.numeric(g_dmpfr[which(g_dmpfr>0)]))),
         max(c(true_g, as.numeric(g_dmpfr)), na.rm=TRUE) )
plot(grid, g_dmpfr, type = "l", col = "red", ylim = ylim, log = "y")
lines(grid, g_d, type = "l")
lines(grid, true_g, col = "blue")

ElliptCopulas documentation built on Sept. 11, 2024, 6:50 p.m.